Finding Surface Area and Volume Lesson 30 Finding Surface Area and Volume

Getting the idea To find surface area of a solid figure, add all the areas of all of its surfaces. (Hence the name, surface area). You can use formulas to find the surface areas of some three-dimensional figures.

Formulas Rectangular Prism SA= 2lw+2wh+2lh l= length; w=width; h=height Triangular Prism SA= 3A+2B A= area of each rectangular face B= area of each triangular base Rectangular Pyramid SA= B + ½ Ps B= area of rectangular base P= perimeter of base s= slant height (height of each triangular face) Tetrahedron (Triangular Pyramid) SA=4A A=area of each triangular face Sphere SA= 4πr2 r=radius of sphere

Example 1 The wooden square pyramid below has a square base with sides 2 centimeters long, and a slant height of 4 centimeters. Find the total surface area. Which formula is the correct formula to use? 4cm 2cm

Continued SA= B + ½ Ps B= area of base= l x w= 2 x 2=4 sq cm P= Perimeter of the base= 2(l+w)= 2(2+2)=8cm Substitute in those values to find the total SA= 4 sq cm + ½(8cm)(4cm) SA= 4sq cm + 16 sq cm SA = 20 sq cm

Example 2 Bradley is going to pant all the surfaces of the wooden pyramid shown in the previous example, except for the square base. When he is finished, how many square centimeters of the pyramid will be painted. What parts are being painted? 4 triangular faces What do we need to find then? A of 1 triangle= ½ bh. How many do we have? So what amount will be painted?

Volume Any prism V= Bh B= area of base, h=height Rectangular Prism V=lwh Any pyramid V=1/3Bh B=area of base, h-height Rectangular pyramid V=1/3lwh Note: Notice that the formula for finding the volume of a pyramid (V=1/3Bh) is 1/3 times the formula for finding the volume of the prism (V=Bh). So, if a pyramid and a prism have the same base and the same height, the volume of the pyramid will be 1/3 the volume of the prism.

More Volume Cylinder V=πr2h R=radius, h=height Cone V= 1/3 πr2h Note: Notice that the formula for finding the volume of a cone is 1/3 times the formula for finding the volume of a cylinder. So, if a cone and a cylinder have the same radius and the same height, the volume of the cone will be 1/3 the volume of the cylinder. Sphere V= 4/3 πr3

Example 3 A cone has a radius of 8cm and a height of 12 cm. What is the volume of the cone? What would be the volume of a cylinder that had a circular base with the same radius and had the same height as the cone shown above? 12cm 8cm

Use the formula to find the volume of the cone 1/3 πr2h V= 1/3 x 3.14 x 82 x 12 V is about 803.84 cu cm Find the volume of a cylinder with the same radius and the same height. What do we know about the relationship between the 2 formulas?

Example 4 Each edge of a cube is 2 cm. Suppose all of the dimensions of this cube were multiplied by a scale factor of 3 to create a bigger cube. Compare the surface areas of the original cube and the enlarged one. Compare the volumes of the original cube and the enlarged one. 2cm 2cm 2cm

Strategy Find the surface area and volume of the original cube. Then enlarge the dimensions and find the surface area and volume of the enlarged cube. Find surface area and volume of the original cube. Enlarge each dimensions by a scale factor of 3. Find the surface area and volume of the enlarged cube. Compare the surface areas. Compare the volumes.